Sulfamic acid

In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ★, on the functions on ℝ²ⁿ, equipped with its Poisson bracket (with a generalization to symplectic manifolds below). This particular star product is also sometimes called Weyl-Groenewold product, as it was introduced by H. J. Groenewold in 1946, in a trenchant appreciation of Weyl quantization —Moyal actually appears not to know about it in his celebrated paper, and in his legendary correspondence with Dirac, as adduced in his biography. (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his flat phase-space quantization picture.)

Definition

The product for smooth functions f and g on ℝ²ⁿ takes the form

$f\star g=fg+\sum _{n=1}^{\mathrm{\infty }}{\hslash }^n{C}_n(f,g)$

where each C_n is a certain bidifferential operator of order n with the following properties. (See below for an explicit formula).

$1.f\star g=fg+\mathcal{𝒪}(\hslash )$

(Deformation of the pointwise product) — implicit in the definition.

$2.f\star g-g\star f=\mathrm{i}\hslash \{f,g\}+\mathcal{𝒪}({\hslash }^2)\equiv \mathrm{i}\hslash \{\{f,g\}\}$

(Deformation of the Poisson bracket, called Moyal bracket.)

$3.f\star 1=1\star f=f$

(The 1 of the undeformed algebra is also the identity in the new algebra.)

$4.\overline{f\star g}=\bar{g}\star \bar{f}$

(The complex conjugate is an antilinear antiautomorphism.)

Note that, if one wishes to take functions valued in the real numbers, then an alternative version eliminates the $\mathrm{i}$ in condition 2 and eliminates condition 4.

If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra A_n, and the two offer alternative realizations of Weyl quantization of the space of polynomials in n variables (or, the symmetric algebra of a vector space of dimension 2n).

To provide an explicit formula, consider a constant Poisson bivector ∏ on ℝ²ⁿ:

$\mathrm{\Pi }=\sum _{i,j}{\mathrm{\Pi }}^{ij}{\partial }_i\wedge {\partial }_j,$

where ∏^ij is a complex number for each i,j.

The star product of two functions $f$ and $g$ can then be defined as

$f\star g=fg+\frac{i\hslash }{2}\sum _{i,j}{\mathrm{\Pi }}^{ij}({\partial }_{i}f)({\partial }_{j}g)-\frac{{\hslash }^2}{8}\sum _{i,j,k,m}{\mathrm{\Pi }}^{ij}{\mathrm{\Pi }}^{km}({\partial }_i{\partial }_{k}f)({\partial }_j{\partial }_{m}g)+\dots$

where ħ is the reduced Planck constant, treated as a formal parameter here.

A closed form can be obtained by using the exponential,

$f\star g=m\circ e^{\frac{i\hslash }{2}\mathrm{\Pi }}(f\otimes g),$

where $m$ is the multiplication map, $m(a\otimes b)=ab$, and the exponential is treated as a power series, $e^A:=1+{\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n!}{A}^n$.

That is, the formula for $C_n$ is

$C_n=\frac{i^n}{2^{n}n!}m\circ {\mathrm{\Pi }}^n.$

As indicated, often one eliminates all occurrences of $\mathrm{i}$ above, and the formulas then restrict naturally to real numbers.

Note that if the functions f and g are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).

On manifolds

On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure constant, by Darboux's theorem; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally, as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a flat symplectic connection.

More general results for arbitrary Poisson manifolds (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula.

Examples

A simple explicit example of the construction and utility of the ★-product (for the simplest case of a two-dimensional euclidean phase space) is given in the article on Weyl quantization: Two Gaussians compose with this ★-product according to a hyperbolic tangent law,

$\exp (-a(x^2+p^2))\star \exp (-b(x^2+p^2))=\frac{1}{1+{\hslash }^{2}ab}\exp (-\frac{a+b}{1+{\hslash }^{2}ab}(x^2+p^2)).$

N.B. Note the classical limit, Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..

Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ★-product.